# Continuous Geometry-Aware Graph Diffusion via Hyperbolic Neural PDE

arXiv: https://arxiv.org/abs/2406.01282 · HTML: https://mlsanity.com/paper/2406.01282/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 56 references that are in the map.
- First posted on arXiv 3 Jun 2024.

## Builds on (most cited first, 30 of 38)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Spatial Temporal Graph Convolutional Networks for Skeleton-Based Action Recognition](https://mlsanity.com/paper/1801.07455/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [DeepGCNs: Can GCNs Go as Deep as CNNs?](https://mlsanity.com/paper/1904.03751/) (2019)
- [Graph Transformer Networks](https://mlsanity.com/paper/1911.06455/) (2019)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Dirichlet Energy Constrained Learning for Deep Graph Neural Networks](https://mlsanity.com/paper/2107.02392/) (2021)

## Built on by (most cited first, 2 of 2)

- [ProGRank: Probe-Gradient Reranking to Defend Dense-Retriever RAG from Corpus Poisoning](https://mlsanity.com/paper/2603.22934/) (2026)
- [Spectral Contraction of Boundary-Weighted Filters on delta-Hyperbolic Graphs](https://mlsanity.com/paper/2506.15464/) (2025)

## Abstract (arXiv)

While Hyperbolic Graph Neural Network (HGNN) has recently emerged as a powerful tool dealing with hierarchical graph data, the limitations of scalability and efficiency hinder itself from generalizing to deep models. In this paper, by envisioning depth as a continuous-time embedding evolution, we decouple the HGNN and reframe the information propagation as a partial differential equation, letting node-wise attention undertake the role of diffusivity within the Hyperbolic Neural PDE (HPDE). By introducing theoretical principles textite.g., field and flow, gradient, divergence, and diffusivity on a non-Euclidean manifold for HPDE integration, we discuss both implicit and explicit discretization schemes to formulate numerical HPDE solvers. Further, we propose the Hyperbolic Graph Diffusion Equation (HGDE) -- a flexible vector flow function that can be integrated to obtain expressive hyperbolic node embeddings. By analyzing potential energy decay of embeddings, we demonstrate that HGDE is capable of modeling both low- and high-order proximity with the benefit of local-global diffusivity functions. Experiments on node classification and link prediction and image-text classification tasks verify the superiority of the proposed method, which consistently outperforms various competitive models by a significant margin.